Lesson TVM xx. Present Value Annuity Due

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1 Lesson TVM xx Present Value Annuity Due This workbook contains notes and worksheets to accompany the corresponding video lesson available online at: Permission is granted for educators and students to make copies and redistribute this document without fee provided the copyright notice and page footer is retained. All other intellectual property rights are reserved by the copyright holder. Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 1 of 26

2 Corrections Subsequent to the production of these videos in July 2010, we re-structured and renumbered them to be consistent with a structure that emerged from our later work in which the last two digits of our numbering scheme represents part 1, part 2, part3, etc. of a lesson. This allows us to break lessons into shorter related subsets easier for you as viewer to manage. We re-edited the videos to refer to the new number structure. However, there may be some references to lesson numbers in the videos that still refer to the old numbering scheme. If so, please alert us so that we can correct that in a future release. For your convenience, the numbering systems are as follows: OLD NUMBER SCHEME NEW NUMBER SCHEME TOPIC TVM TVM xx TVM Part 1 TVM Part 2 Future Value of a Single Sum TVM TVM xx TVM Part 1 TVM Part 2 Future Value of an Ordinary Annuity TVM TVM xx TVM Part 1 TVM Part 2 Present Value of a Single Sum TVM TVM xx TVM Part 1 TVM Part 2 Present Value of an Ordinary Annuity TVM TVM xx TVM Part 1 TVM Part 2 Future Value of an Annuity Due TVM TVM xx TVM Part 1 TVM Part 2 Present Value of an Annuity Due We apologize for any confusion this may cause. Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 2 of 26

3 Time Value of Money - Present Value - Annuity Due - Part 1 [Clip 01] Annuity Due Concept Pre-requisites Understand the concepts of future value of a single sum and present value of a single sum. These are covered in Lesson TVM Present Value Single Sum. We will assume you understand the concepts and problems from this earlier lesson and we will refer to prior examples as we study the present value of an Annuity Due. Recommended Understand the concepts of present value of an ordinary annuity covered in Lesson TVM Present Value Ordinary Annuity. The relationship between an ordinary annuity and an annuity due is explained in this lesson and provides an intuitive understanding that otherwise might be not be understood. However, this relationship is not essential to understanding an annuity due, so you do not have to watch the lesson on ordinary annuities to understand this lesson. Objectives: 1. Annuity Due concept 2. Present value of an Annuity Due formula, tables, financial calculator 3. Solve present value of an Annuity Due problems 4. Solve combination problems involving present value of a single sum and present value of an annuity due Concept of an Annuity Due Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 3 of 26

4 In this lesson we study how to determine the present value of an annuity due. An annuity is a cash payment or a cash receipt that occurs at equal intervals (periods) for a number of intervals (periods). The intervals can be years, semiannual periods, quarterly periods, monthly periods, etc. An annuity amount can be fixed, that is, the same each period, or it can be variable. Our interest in this lesson is a fixed annuity amount, because we have special procedures to determine the present value of a fixed annuity. A variable annuity problem is solved using a combination of fixed annuity and single sum procedures. An annuity due is one in which the cash payment or cash receipt occurs at the beginning of a period. This lesson will deal with that situation. Examples of annuities due include: 1. Lottery winnings in which periodic payments are made at the beginning of each period. 2. Retirement annuity payments in which periodic payments are made at the beginning of each period. 3. Insurance annuity payments in which periodic payments are made at the beginning of each period. 4. Lease or rent payments in which periodic payments are made at the beginning of each period. When the cash payment or cash receipt occurs at the end of each period, it is called an ordinary annuity. We study the present value of an ordinary annuity in Lesson TVM An example of an ordinary annuity is an installment loan in which loan payments occur at the end of each month from the point of view of the inception of the loan. Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 4 of 26

5 Comparison of Time Value of Money Situations Period FV SS PV SS FV OA PV OA FV AD PV AD 0 - Today -PMT 0 -PV 0? -PV 0? -PV 0? or -PV 0 -PMT 1 +PMT 1 End of Period 1 -PMT 1 +PMT 1 -PMT 2 +PMT 2 End of Period 2 -PMT 2 +PMT 2 -PMT 3 +PMT 3 End of Period 3 -PMT 3 +PMT 3 +PMT n +FV n? or +FV n -PMT n-1 +PMT n-1 End of Period N -PMT n +FV n? +PMT n FV n? This table compares the cash flow streams for different time value of money situations. Notice how the column for present value of annuity due (PV AD ) differs from the present value of a single sum (PV SS ) and present value of an ordinary annuity (PV OA ). In the present value of a single sum, we typically know a future amount at the end of some future period, call it PMT n or FV n. We want to know the present value of that amount today at some discount rate, call it r. We are asking How much are we willing to pay today for a future amount? In present value of an ordinary annuity, we know a future amount that occurs at the end of every period from period 1 through period n. We want to know the present value of those amounts today at some discount rate, call it r. We are Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 5 of 26

6 asking How much are we willing to pay today for a stream of future amounts that are the same each period? In present value of an annuity due, we know a future amount that occurs at the beginning of every period from period 1 through period n, which is also to say that the future amount occurs today, and at the end of every period from period 1 through n-1. We want to know the present value of those amounts today at some discount rate, call it r. We are asking How much are we willing to pay today for a stream of future amounts at the beginning of each period starting today that are the same each period? We can also turn this around a bit and ask If we have an amount today, call it PV 0, invested at a certain rate of return, r, what is the amount of the future payments we could receive at the beginning of each period starting today through period n? Another question we might ask is If we have an amount today, call it PV 0, and we want to have a certain periodic payment at the beginning of each period starting today for a certain number of periods n, what rate of return is required on that amount today? A final variation is If we have an amount today, call it PV 0, and we want to have a certain periodic payment at the beginning of each period starting today given a certain rate of return, for how many periods can the payment be made? Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 6 of 26

7 Present Value of an Annuity Due Example Suppose you win a local lottery that will pay you either (1) $10,000 a year for three years starting now, or (2) the cash value of your winnings now assuming a 5% discount rate. How much is the cash value of your lottery winnings? Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 7 of 26

[Clip 02] Contrast an Annuity Due with an Ordinary Annuity [Clip 03] Present Value of an Annuity

[Clip 05] Present Value of an Annuity Due of

10 [Clip 05] Present Value of an Annuity Due of $1 Table See Appendix 2 for the Present Value Tables Used in this clip. [Clip 06] Present Value of an Annuity Due - Using the TI BA II Financial Calculator Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 10 of 26

12 Time Value of Money - Present Value - Annuity Due - Part 2 [Clip 07] Problem 1 Annuity Due Solve for PV Problem 1 Annuity Due Solve for PV (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you buy a factory building under a lease agreement that requires payments of $100,000 per year for 5 years occurring at the beginning of each year. At the end of the lease agreement ownership of the building transfers to your company. This is essentially a financing agreement to buy the factory building and is considered a capital lease. If the implied interest rate of this agreement is 8%, what is the present value of the lease payments (which is also the proper valuation of the acquisition cost of the building)? [Clip 08] Problem 2 Annuity Due Solve for N Problem 2 Annuity Due Solve for N (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you have a lump sum of $220,000 in a retirement fund that earns 4% annually. Assume you are planning to retire immediately and determine you need at least $20,000 a year from the retirement fund starting now. How many years will the retirement fund be able to pay you $20,000 a year? Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 12 of 26

13 [Clip 09] Problem 3 Annuity Due Solve for r Problem 3 Annuity Due Solve for r (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you have a lump sum of $220,000 in a retirement fund. Assume you are planning to retire immediately and determine you need at least $20,000 a year from the retirement fund starting now. What rate of return on the retirement fund is necessary in order for the fund to pay you $20,000 a year for 20 years? [Clip 10] Problem 4 Annuity Due Solve for A Problem 4 Annuity Due Solve for A (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you have a lump sum of $220,000 in a retirement fund earning 5% annually. Assume you are planning to retire immediately. What is the maximum amount your retirement fund can pay you each year for 20 years starting now? [Clip 11] Problem 5 Annuity Due Pension Annuity vs. Lump Sum Analysis Problem 5 Annuity Due Pension Annuity vs. Lump Sum Analysis (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you are retiring at age 65 and have a choice between accepting pension payments for the next 25 years of $50,000 annually starting now, versus a lump sum amount now of a $1,000,000. Assume your discount rate is 7%. Should you take the lump sum amount now? Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 13 of 26

14 [Clip 12] Problem 6 Capital Lease - Combination Single Sum and Annuity Due Problem 6 Capital Lease - Combination Single Sum and Annuity Due (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you are offered the opportunity to buy a new tractor for your farm with a market price of $40,000. The dealer offers to finance the tractor under a lease purchase agreement in which you must make annual payments of $5,000 beginning immediately, for six years with a balloon payment due at the end of six years of $20,000. How much are you paying for the tractor, assuming the appropriate implied interest rate is 5%? [Clip 13] Problem 7 Capital Lease Combination Single Sum and Annuity Due with annual compounding Problem 7 - Capital Lease Combination Single Sum and Annuity Due with annual compounding (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you have the opportunity to lease some factory equipment under a capital lease agreement requiring annual lease payments of $60,000 at the beginning of each year of the lease for five years. At the end of the lease term, an additional payment is required of $90,000 as a guarantee on disposal of the equipment by the leasing company. If the implied interest rate is 9%, what is the present value of the capital lease? Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 14 of 26

15 [Clip 14] Problem 8 Capital Lease Combination Single Sum and Annuity Due with monthly compounding Problem 8 - Capital Lease Combination Single Sum and Annuity Due with monthly compounding (see Appendix 1 for the Solution; see Appendix 2 for the Time Value of Money tables) Suppose you have the opportunity to lease some factory equipment under a capital lease agreement requiring monthly lease payments of $5,000 at the beginning of each month of the lease for two years. At the end of the lease term, an additional payment is required of $10,000 as a guarantee on disposal of the equipment by the leasing company. If the implied interest rate is 12%, what is the present value of the capital lease? Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 15 of 26

16 APPENDIX 1 - SOLUTIONS [Clip 07] Problem 1 Annuity Due Solve for PV SOLUTION: [Clip 08] Problem 2 Annuity Due Solve for N SOLUTION: Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 16 of 26

[Clip 09] Problem 3 Annuity Due Solve for r

for A SOLUTION: Copyright 2011 by Rocky Spears

Combination Single Sum and Annuity Due with monthly compounding SOLUTION:

20 [Clip 13] Problem 7 Capital Lease Combination Single Sum and Annuity Due with annual compounding SOLUTION: [Clip 14] Problem 8 Capital Lease Combination Single Sum and Annuity Due with monthly compounding SOLUTION: Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 20 of 26

21 APPENDIX 2 - FUTURE VALUE AND PRESENT VALUE TABLES Table 1. Future Value of a Single Sum of $1 (Cash Flow or Payment Occurs the Beginning of the Period) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 21 of 26

22 Table 2. Future Value of an Ordinary Annuity of $1 (Cash Flows or Payments Occurs at the End of Each Period) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 22 of 26

23 Table 3. Present Value of a Single Sum of $1 (Cash Flow or Payment Occurs the End of the Period) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 23 of 26

24 Table 4. Present Value of an Ordinary Annuity of $1 (Cash Flows or Payments Occur at the End of Each Period) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 24 of 26

25 Table 5. Future Value of an Annuity Due of $1 (Cash Flows or Payments Occur at the Beginning of the Period) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 25 of 26

26 Table 6. Present Value of an Annuity Due of $1 (Cash Flows or Payments Occur at the Beginning of Each Period) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% Copyright 2011 by Rocky Spears Enterprises LLC, All Rights Reserved Page 26 of 26

Lesson FA xx Capital Budgeting Part 2C

Lesson FA xx Capital Budgeting Part 2C - - - - - - Cover Page - - - - - - Lesson FA-20-170-xx Capital Budgeting Part 2C These notes and worksheets accompany the corresponding video lesson available online at: Permission is granted for educators